Static Games of Incomplete Information
Econ 610 - Game Theory
Agenda
Definitions and Strategies
Bayesian Nash Equilibrium
Example: Cournot Duopoly
Example: Jury Voting
Example: Independent Private Value Auction
Example: Common Value Auction
Incomplete Information
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Complete Information: All information relevant to the game is common
knowledge.
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Incomplete Information: Some payo↵ relevant information is not common
knowledge.
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Examples:
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Cournot game: firm A may not know cost of firm B.
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In-house Auction: bidder may not know valuation of other bidders.
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Online Auction: bidder may not know how many other bidders there are in the game.
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Auction: bidder may not know how much money other bidders have.
Example
Two firms in industry
Firm 2 deciding whether to enter.
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Incomplete Information: Firm 2 doesn’t know firm 1’s building costs.
Enter
Don’t
0 -1
2 0
2 1
3 0
Firm 1 has High Costs
Enter
Don’t
Build
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3 -1
5 0
Don’t
Firm 1 already operating, deciding whether to build new factory.
Build
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Don’t
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2 1
3 0
Firm 1 has Low Costs
Build
0 -1
2 0
2 1
3 0
Firm 1 has High Costs
Enter
Don’t
Build
Don’t
1.5 -1
3.5 0
Don’t
Enter
Don’t
Example
2 1
3 0
Firm 1 has Low Costs
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Firm 1 no longer has dominant strategy with low costs.
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Decision depends on whether or not Firm 2 enters.
Harsanyi Transorfmation
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Harsanyi (1967)
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Haransyi Transformation: Transforms a game of incomplete information into a
game of imperfect information.
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Consists of two steps:
1. Model all asymmetric information about the game as asymmetric information about
how payo↵s depend on actions.
2. Transform asymmetric information about payo↵s into asymmetric information on the
realization of random variables for which the probability distribution is common
knowledge among all players.
Harsanyi Transformation - Step 1
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Possible asymmetric information about the game:
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How many players?
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What actions does each player have?
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How will the outcome depend on actions chosen by players?
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What are players’ preferences over outcomes?
Harsanyi proposed following:
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Is player in game? If not, give only one feasible action, “Out”.
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Is an action feasible? If not, give a very low payo↵ when that action is chosen.
Now all asymmetric information about game is asymmetric information about how
payo↵s depend on actions.
Harsanyi Transformation - Step 2
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Only asymmetric information remaining is how payo↵s depend on actions.
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Players need to form beliefs about which game they are playing.
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Each game is represented with a “state of nature,” ! 2 ⌦.
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Introduce additional player “Nature,” which selects the state of nature.
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The prior belief on the distribution of ⌦ is common knowledge.
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Each player has type, which is private information based on !.
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Ti is the set of possible types for player i.
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Type function ⌧i : ⌦ ! Ti .
Example
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Two states of nature {!1 , !2 } = ⌦.
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Common knowledge prior distribution p (!1 ) = p1 .
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Types:
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⌧1 (!1 ) = t1H and ⌧1 (!2 ) = t1L .
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⌧2 (!1 ) = ⌧2 (!2 ) = t2
High [p1 ]
Build
Enter
(0,-1)
1
x2
N
x1
Low [1
Don’t
p1 ]
Build
1
x3
Don’t
2
2
2
2
x4
x5
x6
x7
Don’t
(2,0)
Enter
(2,1)
Don’t
(3,0)
Enter
(1.5,-1)
Don’t
(3.5,0)
Enter
(2,1)
Don’t
(3,0)
Bayesian Game
Definition (Bayesian Game)
A Bayesian Game consists of
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a finite set of players N
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a finite set of states ⌦
and for each player i 2 N,
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a set Ai of actions
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a finite set Ti of types and a function ⌧i : ⌦ ! Ti .
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a probability measure pi on ⌦ for which pi ⌧i
a payo↵ function ui : S ⇥ T ! R.
1
(ti ) > 0 for all ti 2 Ti .
Beliefs
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In general games, every player may have private information.
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My private information may tell me something about your private information.
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Example: in an auction our values are likely correlated.
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Player i’s posterior belief about other’s types given ti is given using Bayes’ rule,
(
pi (!)
! 2 ⌧i 1 (ti )
pi (⌧i 1 (ti ))
pi (!|ti ) =
0
else
where ⌧i
1
(ti ) is the set of states of the world that lead to ti .
Strategies
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A strategy si : Ti ! Ai is a contingency plan that gives an action for each
possible type.
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Example: if |A1 | = 7 and |T1 | = 4, how many pure-strategies does player 1 have?
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Types of strategies:
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Pooling Strategy: si (t) = x for all t 2 Ti .
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Separating Strategy: si (t) 6= si (t 0 ) for all t, t 0 2 Ti such that t 6= t 0 .
Bayesian Nash Equilibrium
Definition (Bayesian Nash Equilibrium)
A strategy profile s ⇤ = (s1⇤ , . . . , sn⇤ ) is a pure-strategy Bayesian Nash equilibrium if for
each i 2 N and for each ti 2 Ti ,
X
si⇤ (ti ) = argmax
pi (!|ti ) ui ai , s ⇤ i (t i ) , !
ai 2Ai
!2⌦
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Interpretation: Given my beliefs, my strategy must be a best response to other
players strategies.
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Existence: a Bayesian Nash equilibrium exists in any finite Bayesian Game.
Cournot Duopoly with Incomplete Information
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Two firms (Firm A and Firm B)
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Firms select quantities qA 2 [0, 1) and qB 2 [0, 1).
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Market demand,
P (qA , qB ) = 2
qA
qB
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Firm A has cost cA = 1.
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Firm B has costs cB = 3/4 and cB = 5/4 with equal probability.
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Firm B know cB , but Firm A does not.
As Bayesian Game
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Players: N = {FA , FB }.
States: ⌦ = {L, H}.
Types:
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Type functions:
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⌧A (L) = ⌧A (H) = tA .
⌧B (L) = tBL and ⌧B (H) = tBH .
Actions:
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TA = {tA }
TB = tBL , tBH
AA = [0, 1)
AB tBL = AB tBH = [0, 1)
Prior probability: pi (L) = pi (H) = 1/2 for i = A, B.
Payo↵s:
⇡A (qA , qB , !) = qA (2 qA qB ) qA
⇢
qB (2 qA qB ) 34 qB if ! = L
⇡B (qA , qB , !) =
qB (2 qA qB ) 54 qB if ! = H
Unanimous Juries
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Blackstone’s Formulation:
“better that ten guilty persons escape than that one innocent su↵er”
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The United States Supreme Court ruled in Apodaca v. Oregon that the Sixth
Amendment to the Constitution mandates unanimity for a guilty verdict in a
federal court jury trial.
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Feddersen and Pesendorfer (1998): Do unanimous juries make it less likely that an
innocent person is convicted?
Jury Voting Model
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n jurors
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Defendant is either guilty (G ) or innocent (I ) with equal probability.
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Each juror receives a signal g or i such that,
P (g |G ) = P (i|I ) = p with p 2 (.5, 1)
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Jurors vote simultaneously to convict (C ) or acquit (A).
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ˆ then the
If the number of votes to convict (#C ) is greater than or equal to k,
defendant is convicted, otherwise defendant is acquitted.
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Preferences:
u (A|I ) = 0
u (C |G ) = 0
u (A|G ) = (1
u (C |I ) = q
q)
As Bayesian Game
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Players: N = {1, . . . , n}
States: ⌦ = {(X , s1 , . . . , sn ) |X 2 {G , I } and si = {g , i}}
Types: Ti = {g , i}
Type functions: ⌧i (!) = si for ! = (X , s1 , . . . , sn ).
Actions: Ai (g ) = Ai (i) = {A, C }.
Prior probability:
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Let k = # correct signals in !.
Let n k = # incorrect signals in !.
pi (!) =
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Payo↵ (let #C = |{ai |ai = C }|).
8
>
>
>
<
u (a1 , . . . , an , !) =
>
>
>
:
0
1 k
p (1
2
if
(1 q) if
q
if
0
if
p)
n k
kˆ > #C
kˆ > #C
#C kˆ
#C kˆ
and
and
and
and
I 2!
G 2!
I 2!
G 2!
Types of Voting
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Informative Voting: Juror votes A with signal i and votes C with signal g .
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Strategic Voting: Juror evaluates payo↵s and may vote against signal.
Case #1: Informative Voting
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Assume that all voters vote informatively:
Probability a convicted defendant is guilty is,
P (G |C ) =
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P (C |G ) P(G )
=
P (C |I ) P(I ) + P (C |G ) P(G )
(1
pn
p)n + p n
! 1
n!1
Probability of convicting innocent (Type I) lowest with unanimity:
P Unan (C |I ) < P k (C |I ) for all k < n
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Probability of acquitting guilty (Type II) is highest with unanimity:
P Unan (A|G ) > P k (A|G ) for all k < n
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With informative voting, unanimity minimizes Type I error at the cost of Type II
error.
Informative voting is typically not equilibrium behavior.
Case #2: Strategic Voting
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Let the posterior probability of being guilty conditional on k out of n signals being
g be,
(k, n) =
=
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Note: if
P (g |G )k P (i|G )n
P (g |G )k P (i|G )n
p k (1
p k (1
p)n
k
k
p)n
+ (1
+ P (g |I )k P (i|I )n
k
k
p)k p n
k
(k, n) > q, juror prefers to convict.
u(C ) = u (C |G ) + (1
) u (C |I ) =
q (1
u(A) = u (A|G ) + (1
) u (A|I ) =
(1
U(C ) > U(A) () q <
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k
q can be though of as level of reasonable doubt
)
q)
Case #2: Strategic Voting
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Assumption: there exists k ⇤ (with 1  k ⇤  n) such that,
(k ⇤
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1, n)  q <
If no k ⇤ exists, then either:
1. Even if all signals are i, would always convict.
q<
(0, n)
2. Even if all signals are g , would never convict.
(n, n) < q
(k ⇤ , n)
Pivotal Voter
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Let Ui be expected payo↵.
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Let ui be payo↵ for specific outcome (0, q, (1
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Payo↵ for voting C and A are,
Ui (C ,
i , !|ti )
=
X
q)).
pi (!|ti ) ui (C ,
i
(t i ) , !)
pi (!|ti ) ui (A,
i
(t i ) , !)
!2⌦
Ui (A,
i , !|ti )
=
X
!2⌦
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Let ⌦0 ✓ ⌦, such that ! 2 ⌦0 if and only if,
ui (C ,
I
i
(t i , !)) = ui (A,
Players only need to consider ! 2 ⌦ \ ⌦0 .
i
(t i , !))
Case #2: Strategic Voting
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Pivotal Voter: a juror is said to be pivotal if their vote will change the outcome.
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With strategic voting, juror only analyzes decision based on being pivotal.
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Case 2.1: k ⇤ = n )
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(n
1, n)  q <
(n, n).
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Juror’s posterior belief about guilt is only strong enough if all signals are guilty.
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In this case, informative voting is an equilibrium.
Case 2.2: k ⇤ < n ) q <
(n
1, n).
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Even if juror receives i, posterior belief about guilt may be strong enough to vote
convict.
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In this case, informative voting is not an equilibrium.
Case #2: Strategic Voting
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What are the equilibria when k ⇤ < n?
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There are many (example: always acquit).
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Let (s) be the probability of voting C given signal s.
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Focus on symmetric Nash equilibrium ( (i), (g )).
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Define
G
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= (g )p + (i) (1
= (g ) (1
p) = P (juror votes convict|G )
p) + (i)p = P (juror votes convict|I )
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Strategy is said to be responsive (depends on signal) if
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Focus on responsive symmetric Nash equilibrium.
G
6=
I.
Case #2: Strategic Voting
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When q <
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So any responsive equilibrium must be in mixed strategies.
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Equilibrium must be (i) 2 (0, 1) and (g ) = 1.
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In equilibrium, juror must be indi↵erent between C and A when they get signal i.
(n
1, n), informative voting is not an equilibrium.
Case #2: Strategic Voting
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When q <
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So any responsive equilibrium must be in mixed strategies.
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Equilibrium must be (i) 2 (0, 1) and (g ) = 1.
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In equilibrium, juror must be indi↵erent between C and A when they get signal i.
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Equilibrium:
(n
1, n), informative voting is not an equilibrium.
⇤
⇤
(i) =
⇣
p
(g ) = 1
⌘1/n
(1 q)(1 p)
qp
⇣
(1 q)(1 p)
qp
1
p
⌘1/n
1
(1
p)
(1
p)
Case #2: Strategic Voting
⇤ (i),
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For the equilibrium (
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Denote Type I error by,
⇤ (g )):
II (p, q, n) = ( I )n = P (C |I )
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Denote Type II error by,
IG (p, q, n) = 1
(
G)
n
= P (A|G )
Case #2: Strategic Voting
Proposition
Assume k ⇤ < n and q > 1 p. The strategy given above is the unique responsive
symmetric equilibrium for the unanimity rule. Moreover, (i) ! 1 as n ! 1, and
lim lI (p, q, n) =
n!1
lim lG (p, q, n) = 1
n!1
✓
(1
✓
(1
q) (1
qp
q) (1
qp
p)
◆p/(2p
p)
◆1
1)
p/(2p 1)
If q  1 p, then there is no responsive equilibrium. In this case (i) = 1 is an
equilibrium, and lI (p, q, n) = 1, lG (p, q, n) = 0.
Case #2: Strategic Voting
Case #2: Strategic Voting
Case #2: Strategic Voting
Case #2: Strategic Voting
Auctions
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Di↵erent types of auctions:
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First price - highest bidder wins, pays bid.
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Second price - highest bidder wins, pays second highest bid.
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All-Pay - highest bidder wins, everyone pays their bid.
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Dutch - start with high-price, gradually decrease until someone accepts.
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English - start with low-price, gradually increase until only one person left.
Which one is best?
Independent Private Value Auction Model
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Risk neutral seller that have $0 value for item.
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N risk-neutral bidders i = 1, . . . , n.
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Buyer i has value vi drawn from Fi (x) on [0, 1] with density fi (x).
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Buyer’s values are mutually independent.
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Bidding function bi : [0, 1] ! R+ .
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Payo↵s (first price):
ui (vi , v i ) =
⇢
vi
b (vi ) if vi > v
0
else
i
As Bayesian Game
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Players: N = {1, . . . , n}
States: ⌦ = [0, 1]n
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Types: Ti = [0, 1]
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Type function: ⌧i (!) = vi .
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Actions: Ai = [0, 1) for all vi for all i.
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Prior probability
pi (!) = f1 (v1 ) f2 (v2 ) · · · fn (vn )
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Payo↵:
ui (a1 , . . . , an , !) =
⇢
vi
ai
0
if vi > vj 8j 6= i 2 N
else
Further Assumptions
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Assume fi (v ) = f (v ) = 1 for all i.
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Focus on bidding functions bi (v ) such that:
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bi (v ) is strictly increasing
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bi (v ) = bj (v ) (symmetric bidding functions)
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Let b ⇤ represent the equilibrium bidding function.
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Questions:
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What is the equilibrium bidding function for a first-price auction?
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What is the sellers revenue from a first-price auction?
Common Value Auction Model
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Risk neutral seller that has $0 value for item.
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N risk-neutral bidders i = 1, . . . , n.
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All buyers have a common value for the item V .
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Bidder i has a signal vi , which in someway is related to the true value,
P
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V =
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vi = V + "i where "i ⇠ U [ 1, 1].
i2N
vi .
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Bidding function bi : [0, 1] ! R+ .
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Payo↵s (first price):
ui (vi , v i ) =
⇢
V
b (vi ) if vi > v
0
else
i
Common Value Auction: Example #1
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Two bidders {A, B} bidding on oil field.
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Two parts of field A and B.
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Bidder i gets signal si telling value of part i.
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Each part is worth either $0 or $3 million.
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Oil field has common value
Oil Field
V = sA + sB
Part A
Part B
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E (sA ) = 1.5.
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E (sB ) = 1.5.
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E (V ) = 3.
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Bidders participate in first price auction for the oil field.
Common Value Auction: Example #2
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N bidders i = 1, . . . , n.
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All bidders have common value v .
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Bidders only receive signal of true value,
si = v + "i
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"i ⇠ U [ 1, 1].
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F (") =
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Bidders then participate in first-price auction.
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Suppose bidders use linear bidding function bi (vi ) = mvi .
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How low does m have to be to avoid winner’s curse?
"+1
2 .